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Quasilinearization approach for solving Volterra's population model

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The study used the Quasilinearization method to solve Volterra's model for popu-lation growth of a species within a closed system is proposed. This model is a nonlinear integro-differential where the integral term represents the effect of toxin. First we convert this model to a nonlinear ordinary differential equation, then approximate the solution of this equation by treating the nonlinear terms as a perturbation about the linear ones. Finally we compare this method with the other methods and come to the conclusion that the Quasilinearization method gives excellent results.
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Appl. Comput. Math., V.9, N.1, 2010, pp.
QUASILINEARIZATION APPROACH FOR SOLVING VOLTERRA’S
POPULATION MODEL
K. PARAND1, M. GHASEMI2, S. REZAZADEH2, A. PEIRAVI2, A. GHORBANPOUR2,
A. TAVAKOLI GOLPAYGANI3
Abstract. The study used the Quasilinearization method to solve Volterra’s model for popu-
lation growth of a species within a closed system is proposed. This model is a nonlinear integro-
differential where the integral term represents the effect of toxin. First we convert this model
to a nonlinear ordinary differential equation, then approximate the solution of this equation by
treating the nonlinear terms as a perturbation about the linear ones. Finally we compare this
method with the other methods and come to the conclusion that the Quasilinearization method
gives excellent results.
Keywords: Volterra’s Population Model, Quasilinearization, Integro-differential, Barycentric.
AMS Subject Classification:
1. Introduction
The Quasilinearization method was developed many years ago in the theory of linear pro-
gramming by Bellman and Kalaba [1, 5] as a generalization of the Newton Raphson method
[4, 11] to solve the systems of nonlinear ordinary and partial differential equations. Modern
developments and applications of the QLM method to different fields are given in a monograph
[6]. This approach is applicable to a general nonlinear nth order ordinary or partial differential
equation in N-dimensional space, and to the complicated nonlinear two-point boundary condi-
tion [1, 5]. To reach convergence, the simple guess of the zeroth iteration being equal to zero or
to one of the boundary conditions, is usually enough [7]. This approach was applied for solving
nonlinear Calogero equation in a variable phase approach to quantum mechanics. The results
were compared with those of perturbation theory and the exact solutions. In this method iter-
ations yield rapid quadratic convergence and often monotonicity. In [8] the Quasilinearization
method was applied to solve different nonlinear ordinary differential equations in physics, such
as Lane-Emden, Thomas-Fermi, Duffing and Blasius equations. They have shown this method
gives excellent results. These equations contain not only quadratic nonlinear terms but also
various other forms of nonlinearity of the higher derivatives. It was shown that a small number
of the QLM iterations yield fast convergence and uniformly stable numerical results.
The Volterra’s model for population growth of a species within a closed system is given in
[15, 12, 14] as
dp
d˜
t=ap bp2cp
˜
t
Z
0
p(x)dx, p(0) = p0,(1)
1Department of Computer Science , Shahid Beheshti University, Tehran, Iran
2Department of Applied Mathematics, Shahid Beheshti University, Tehran, Iran
3Department of Medical physics and Biomedical Engineering, Shiraz University of Medical Sciences,
Shiraz, Iran
Manuscript received 20 September 2008.
1
2 APPL. COMPUT. MATH., V.9, N.1, 2010
where a > 0 is the birth rate coefficient, b > 0 is the crowding coefficient, and c > 0 is the
toxicity coefficient. The coefficient cindicates the essential behavior of the population evolution
before its level falls to zero in the long term. p0is the initial population, and p=p(˜
t) denotes
the population at time ˜
t. This model is a first-order integro-ordinary differential equation where
the term cp
˜
t
R
0
p(x)dx represents the effect of toxin accumulation on the species. we apply scale
time and population by introducing the nondimensional variables
t=˜
tc
b, u =pb
a(2)
to produce the nondimentional problem
κdu
dt =uu2u
t
Z
0
u(x)dx, u(0) = u0,(3)
where u(t) is the scaled population of identical individuals at time t, and κ=c/(ab) is a
prescribed nondimensional parameter. The only equilibrium solution of (3) is the trivial solution
u(t) = 0 and the analytical solution [13]
u(t) = u0exp(1
κ
t
Z
0
[1 u(τ)
τ
Z
0
u(x)dx]),
shows that u(t)>0 for all tif u0>0. In [13], the singular perturbation method for Volterra’s
population model is considered. It is shown in [13] that for the case κ¿1, where populations are
weakly sensitive to toxins, a rapid rise occurs along the logistic curve that will reach a peak and
then is followed by a slow exponential decay. And, for κlarge, where populations are strongly
sensitive to toxins, the solutions are proportional to sech2(t).In [15], the series solution method
and the decomposition method are implemented independently to (3) and to a related nonlinear
ordinary differential equation. Furthermore, the Pade approximations are used in the analysis
to capture the essential behavior of the populations u(t) of identical individuals. He compared
the approximation of umax and exact value of umax for different κ.
The solution of (1) has been of considerable concern. Although a closed form solution has
been achieved in [14, 13], but it was formally shown that the closed form solution cannot lead to
any insight into the behavior of the population evolution [15]. Some approximate and numerical
solutions for Volterra’s population model have been reported. In reference [12], the successive
approximations method was offered for the solution of (3), but was not implemented. In this case
the solution u(t) has a smaller amplitude compared to the amplitude of u(t) for the case κ¿1.
In [14] , several numerical algorithms namely the Euler method, the modified Euler method, the
classical fourth-order Runge-Kutta method and Runge-Kutta-Fehlberg method for the solution
of (3) are obtained. Moreover,a phase-plane analysis is implemented. In [14], the numerical
results are correlated to give insight on the problem and its solution without using perturbation
techniques. However, the performance of the traditional numerical techniques is well known
in that it provides grid points only, and in addition, it requires large amounts of calculations.
In [9, 10] applied spectral method to solve Volterra’s population on a semi-infinite interval.
This approach is based on a Rational tau method. They obtained the operational matrices of
derivative and product of rational functions and reduced the solution of this problem to the
solution of algebraic equations systems.
As one more step in this direction, We use Quasilinearization approach for solving Volterra’s
population model and then we apply Baricentric Lagrange Interpolation with xj’s chebyshev
K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...3
points of the first kind. on the other hand, comparison of results obtained by this method and
others shows that this method provides more accurate and numerically stable solution than
those obtained by the other methods.
The paper is arranged as follows: in section 2 we describe the Quasilinearization method. In
this section the Volterra’s population model is considered. This equation is first converted to
an equivalent nonlinear ordinary differential equation. Then we apply our method to solve this
ODE. In this section, the proposed method is used for the numerical examples, and a comparison
is made with the existing methods in the literature. Then, the results and numerical tables are
stated. The Finding of the study are summarized in the last section.
2. The Quasilinearization method (QLM)
The Quasilinearization method (QLM) is an application of the Newton-Raphson-Kantrovich
approximation method in function space [1]. This method is applied to solve a nonlinear nth
order ordinary or partial differential equation in N-dimensions as a limit of a sequence of linear
differential equations. The idea and advantage of the method is based on the fact that linear
equations can often be solved analytically or numerically using superposition principle while
there are no useful techniques for obtaining the general solution of a nonlinear equation in terms
of a finite set of particular solutions. As we discussed in the introduction, we limit ourselves here
to a nonlinear ordinary differential equation in one variable on the interval [0,b] which could be
infinite:
L(n)y(x) = f(y(x), y(1)(x), ...., y(n1)(x), x),(4)
with n boundary conditions
hi(y(0), y(1)(0), ..., y(n1) (0)) = 0, i = 1, .., d (5)
and
hi(y(b), y(1)(b), ..., y(n1)(b)) = 0, i =d+ 1, .., n, (6)
here L(n)is the linear nth order ordinary differential operator, fand h1, h2, ..., hnare the non-
linear functions of y(x) and its (n-1) derivatives are y(j)(x)j= 1, ..., n 1. The QLM
prescription [1, 5, 7] determines the (r+ 1)th iterative approximation yr+1(x) to the solution of
(4) as a solution of the linear differential equation:
L(n)yr+1(x) = f(yr(x), y(1)
r(x), ...., y(n1)
r(x), x)+ (7)
+
n1
X
j=0
(y(j)
r+1(x)y(j)
r(x))fy(j)(yr(x), y(1)
r(x), ...., y(n1)
r(x), x),
where y0
r(x) = yr(x), with linearized two-point boundary conditions
n1
X
j=0
(y(j)
r+1(0) y(j)
r(0))hky(j)(yr(0), y(1)
r(0), ...., y(n1)
r(0),0) = 0, k = 1, .., d (8)
and
n1
X
j=0
(y(j)
r+1(b)y(j)
r(b))hky(j)(yr(b), y(1)
r(b), ...., y(n1)
r(b), b) = 0, k =d+ 1, .., n. (9)
Here the functions fy(j)=∂f
∂yjare functional derivatives of the functionals
f(y(x), y(1)(x), ...., y(n1)(x), x)
4 APPL. COMPUT. MATH., V.9, N.1, 2010
and
hk(y(x), y(1)(x), ...., y(n1)(x), x)
respectively. The zeroth approximation y0(t) is chosen from mathematical or physical consid-
erations. Where δyr+1 (x) and yr+1(x) are the differences between two subsequent iterations
and between the exact solution and the rth iteration, respectively:
δyr+1(x) = yr+1(x)yr(x),
yr+1(x) = y(x)yr(x).
In [1] proved kyr+1(x)k≤ kkyr(x)k2that showed the difference between the exact solution
and the rth iteration is decreasing quadratically and δyr+1(x) for an arbitrary l < r satisfied the
following inequality:
kδyr+1(x)k≤ (kkδyl(x)k)2rl/k,
or for l= 0, they related the (n+ 1)th order result to the 1st iterate by
kδyn+1(x)k≤ (kkδy1(x)k)2n/k.
Therefore the convergence depends on the quantity q1=kky1y0k.Where the zeroth
iteration y0(x) is chosen from physical and mathematical considerations.
3. Solving Volterra’s population model
3.1. Converting Volterra’s Population Model to a Nonlinear ODE Equation. In this
section we convert Volterra’s population model (3) to an equivalent nonlinear ordinary differen-
tial equation. Let
y(x) =
x
Z
0
u(t)dt. (10)
This leads to
y0(x) = u(x),(11)
y00(x) = u0(x).(12)
Inserting (10)-(12) into (1) yields the nonlinear differential equation
κy00 (x) = y0(x)(y0(x))2y(x)y0(x),(13)
with the initial conditions
y(0) = 0,
y0(0) = u0.
that are obtained by using (10) and (12) respectively.
K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...5
3.2. Solution of Volterra’s Population Model. In the previous section we converted Volterra’s
Population model to a nonlinear ODE. In this section we apply the QLM procedure to solve
this model which is reduced to setting
κy00
n+1(x) = f(yn(x), y0
n(x)) + (yn+1(x)yn(x))fyn+ (y0
n+1(x)y0
n(x))fy0
n,
where f=y0(x)(y0(x))2y(x)y0(x). The (n+ 1)th iteration for volterra’s population model
has the following form:
κy00
n+1(x) = y0
n(x)(y0
n(x) + yn(x)yn+1(x)) + y0
n+1(x)(1 2y0
n(x)yn(x)),
yn+1(0) = 0, y0
n+1(x) = 0.1,(14)
for κ= 0.02,0.04,0.1,0.2 and 0.5, where yn(x) is a previous iteration which is considered to be
a known function. We solve (14) numerically using Maple software and then apply Barycentric
Lagrange Interpolation to approximate yn+1(x) as the following [2, 3]:
yn+1(x) =
m
P
j=0
wj
(xxj)fj
m
P
j=0
wj
(xxj)
,
where xj’s are Chebyshev points of the first kind that are given by:
xj= cos (2j+ 1)π
2m+ 2 , j = 0, .., m.
In this case, after cancelling factors independent of jwe have:
wj= (1)jsin (2j+ 1)π
2m+ 2
and fj’s are the solutions of (14) in xj’s. The initial guess, satisfying the boundary conditions
at zero, was suitably made for different values of κ.
3.3. Illustrative example. We apply the presented method in this paper to examine the math-
ematical structure of u(t). In particular, we seek to study the rapid growth along the logistic
curve that will reach a peak, then followed by the slow exponential decay where u(t)0 as
t . The mathematical behavior defined in this way was introduced by [12] and justified by
[13] by using singular perturbation methods for the inner and outer solutions. Further, these
properties were also confirmed by [14] upon using a phase plane analysis and [15] by using Pade
approximations. We apply the method presented in this paper and solved (3) where u0= 0.1
and κ= 0.02,0.04,0.1,0.2, and 0.5. In Table 1, we compared the results of QLM, method and
exact values of umax for κ= 0.02,0.04,0.1,0.2 and 0.5. In this table muis the minimum QLM
iteration number required in case that the absolute value of differences among the successive
iterations is less than (105).
6 APPL. COMPUT. MATH., V.9, N.1, 2010
Table 1. A comparison of the presented method(QLM) and the exact values for umax
κ muQLM Exact umax
0.02 5 0.92343412 0.92342717
0.04 4 0.87381153 0.87371998
0.1 7 0.76974153 0.76974149
0.2 3 0.65905030 0.65905038
0.5 8 0.48519041 0.48519030
In Fig.1 below is shown the results of QLM calculations for κ= 0.02,0.04,0.1,0.2 and 0.5.
0 1 2 3 4 5 6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k=0.02
k=0.04
k=0.1
k=0.2
k=0.5
Figure 1. The results of QLM calculations for κ= 0.02,0.04,0.1,0.2 and 0.5.
In addition, Fig.1 formally shows the rapid rise along the logistic curve followed by the slow ex-
ponential decay after reaching the maximum point.
In Table 2, the resulting values using the present method together with the results obtained
in [9, 10, 15] and the exact values are presented.
Table 2. A comparison of the method in [9, 10, 15] and the presented method with the exact values for umax.
κMethod in [9] Method in [10] Method in [15] QLM Exact umax
0.02 0.923327 0.923463 0.90383805 0.92343412 0.92342717
0.04 0.873605 0.873708 0.86124018 0.87381153 0.87371998
0.1 0.769623 0.769734 0.76511308 0.76974153 0.76974149
0.2 0.658872 0.659045 0.65791231 0.65905030 0.65905038
0.5 0.485076 0.485188 0.48528235 0.48519041 0.48519030
K. PARAND, M. GHASEMI, S. REZAZADEH, A. PEIRAVI, A. GHORBANPOUR, A. TAVAKOLI GOLPAYGANI: ...7
4. Concolusion
The Quasilinearization method(QLM)is very easy to apply and this method treats the non-
linear terms as perturbation about the linear ones and is not based,unlike perturbation theories,
on the existence of some kind of small parameters.
We used QLM for solving Volterra’s Population Model that in each stage,we solved yn(x) nu-
merically and applied Barycentric Lagrange Interpolation to approximate yn+1(x) with Cheby-
shev points. This interpolation is very good and accurate.
In the end we Compared this method with the other methods and found that the answers
were very accurate, numerically stable and converged fast.
We used an illustrative example to demonstrate the application of this method, and came to
satisfactory results.
References
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[2] Berrut, J.P. Rational Functions for Guaranteed and Experimentally Well-conditioned Global Interpolation,
Comput. Math. Applic., V.15, 1988, pp.1-16.
[3] Berrut, J.P., Trefethen, L.N. Barycentric Lagrange Interpolation, SIAM Rev., V.46, 2004, pp.501-517.
[4] Conte,S.D., Boor,C.de. Elementary Numerical Analysis, McGraw-Hill international editions, 1981.
[5] Kalaba, R. On Nonlinear Differential Equations, The Maximum Operation and Monotone Convergence., J.
Math. Mech., V.8, 1959.
[6] Lakshmikantham, V., Vatsala, A.S. Generalized Quasilinearization for Nonlinear Problems, Mathematics and
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[8] Mandelzweig, V.B., Tabakin, F. Quasilinearization Approach to Nonlinear Problems in Physics with Appli-
cation to Nonlinear ODEs, Comput. Phys. Comm., V.141, 2001, pp.268-281.
[9] Parand, K., Razzaghi, M. Rational Chebyshev Tau Method for Solving Volterra’s Population Model, Applied
Math. Comput., V.14, 2004, pp.893-900.
[10] Parand, K., Razzaghi, M. Rational Legendre Approximation for Solving some Physical Problems on Semi-
infinite Intervals, Physica scripta., V.69, 2003, pp.353-357.
[11] Ralston, A., Rabinowitz, P. A First Course in Numerical Analysis, McGraw-Hill international editions, 1988.
[12] Scudo, F.M., Volterra and Theoretical Ecology, Theoretical Population Biology, V.2, 1971.
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Mathematics, SIAM, Philadelphia, PA, 1989.
[14] TeBeest, K.G. Numerical and Analytical Solutions of Volterra’s Population Model, SIAM Rev., V.39, N.3,
1997, pp.484-493.
[15] Wazwaz, A.M. Analytical Approximations and Pade Approximants for Volterra’s Population Model, Applied
Math. Comput., V.100, 1999, pp.13-25.
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The proof of the convergence of the quasilinearization method of Bellman and Kalaba, whose origin lies in the theory of linear programming, is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems. This powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones, and is not based, unlike perturbation theories, on existence of some kind of small parameter. The general properties of the method, particularly its uniform and quadratic convergence, which often also is monotonic, are analyzed and verified on exactly solvable models in quantum mechanics. Namely, application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions. The method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist. © 1999 American Institute of Physics.
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A numerical technique for solving some physical problems on a semi-infinite interval is presented. Two nonlinear examples are proposed. In the first example the Volterra's population model growth is formulated as a nonlinear differential equation, and in the second example the Lane–Emden nonlinear differential equation is considered. The approach is based on a rational Legendre tau method. The operational matrices of derivative and product of rational Legendre functions are presented. These matrices together with the tau method are utilized to reduce the solution of these physical problems to the solution of systems of algebraic equations. The method is easy to implement and yields very accurate results.
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In this paper, an analytic approximation for Volterra's model for population growth of a species in a closed system is presented. The nonlinear integro-differential model includes an integral term that characterizes accumulated toxicity on the species in addition to the terms of the logistic equation. The series solution method and the decomposition method are implemented independently to the model and to a related ODE. The Padé approximants, that often show superior performance over series approximations, are effectively used in the analysis to capture the essential behavior of the population u(t) of identical individuals.
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Polynomial interpolation is known to be ill-conditioned if the interpolating points are not chosen in special ways; classical rational interpolation can give better results, but does not work in all cases and the corresponding functions can show poles in the interval of interpolation. We present here rational functions which guarantee well-conditioned interpolation on a real interval or a circle and cannot have any poles there. They can be evaluated at least as efficiently as the corresponding interpolation polynomials and the accuracy of their approximation to a given function often compares favorably with that of spline interpolants.
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An approximate method for solving Volterra’s population model for population growth of a species in a closed system is proposed. Volterra’s model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. The approach is based on a rational Chebyshev tau method. The Volterra’s population model is first converted to a nonlinear ordinary differential equation. The operational matrices of derivative and product of rational Chebyshev functions are presented. These matrices together with the tau method are then utilized to reduce the solution of the Volterra’s model to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results.